WHO’S AFRAID OF MATHEMATICAL DIAGRAMS?

Abstract

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions.   Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs.  In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability.  I then consider whether diagrams can be essential to the proofs in which they appear.@font-face{font-family:"Cambria Math";panose-1:2 4 5 3 5 4 6 3 2 4;mso-font-charset:0;mso-generic-font-family:roman;mso-font-pitch:variable;mso-font-signature:-536870145 1107305727 0 0 415 0;}@font-face{font-family:Calibri;panose-1:2 15 5 2 2 2 4 3 2 4;mso-font-charset:0;mso-generic-font-family:swiss;mso-font-pitch:variable;mso-font-signature:-536859905 -1073697537 9 0 511 0;}p.MsoNormal, li.MsoNormal, div.MsoNormal{mso-style-unhide:no;mso-style-qformat:yes;mso-style-parent:"";margin:0in;line-height:200%;mso-pagination:widow-orphan;font-size:12.0pt;font-family:"Calibri",sans-serif;mso-fareast-font-family:Calibri;}.MsoChpDefault{mso-style-type:export-only;mso-default-props:yes;font-family:"Calibri",sans-serif;mso-ascii-font-family:Calibri;mso-fareast-font-family:Calibri;mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri;}.MsoPapDefault{mso-style-type:export-only;line-height:200%;}div.WordSection1{page:WordSection1;}